The Antarctic Ice Sheet is surrounded by large floating ice shelves that spread under their own weight into the ocean. Ice shelf rigidity depends on ice temperature and fabrics, and is influenced by ice flow and the delicate balance between bottom and surface accumulation. Here, we use an inverse control method to infer the rigidity of the Ronne Ice Shelf that best matches observations of ice velocity from satellite radar interferometry. Ice rigidity, or flow law parameter B, is shown to vary between 300 and 900 kPa a1/3. Ice is softer along the side margins due to frictional heating, and harder along the outflow of large glaciers, which advect cold continental ice. Melting at the bottom surface of the ice shelf increases its rigidity, while freezing decreases it. Accurate numerical modelling of ice shelf flow must account for this spatial variability in mechanical characteristics.